Yang–Mills–Higgs Fields

Abstract

Yang–Mills equations, originally derived for the isospin gauge group SU(2), provide the first example of gauge field equations for a non-Abelian gauge group. This gauge group appears as an internal or local symmetry group of the theory. In fact, the theory can be extended easily to include the other classical Lie groups as gauge groups. Historically, the classical Lie groups appeared in physical theories, mainly in the form of global symmetry groups of dynamical systems. Noether’s theorem established an important relation between symmetry and conservation laws of classical dynamical systems. It turns out that this relationship also extends to quantum mechanical systems. Weyl made fundamental contributions to the theory of representations of the classical groups [401] and to their application to quantum mechanics. The Lorentz group also appears first as the global symmetry group of the Minkowski space in the special theory of relativity. It then reappears as the structure group of the principal bundle of orthonormal frames (or the inertial frames) on a space-time manifold M in Einstein’s general theory of relativity. In general relativity a gravitational field is defined in terms of the Lorentz metric of M and the corresponding Levi-Civita connection on M. Thus, a gravitational field is essentially determined by geometrical quantities intrinsically associated with the space-time manifold subject to the gravitational field equations. This geometrization of gravity must be considered one of the greatest events in the history of mathematical physics.